Can a black hole be magnetic if the mass used to form it came from magnetic elements? By magnetic, I mean that it would react strongly to the presence of an external magnetic field. The concept of a spinning black hole creating a magnetic field is unrelated to this question.
My guess is no, because:


*

*the extreme forces compressing the atoms destabilizes and destroys their structure to the point that matter inside a black hole cannot even manifest the emergent properties of combining quarks.

*information inside the black hole cannot leave it, so it would be impossible to emit a magnetic field due to the matter inside the black hole, even if the matter magic-physics-ally could stay in the form of the atoms used to create it.
 A: Not Sure what you mean by 'react strongly'. As quoted by StephenG, Kerr-Newman black holes have charges beyond angular momentum and interact with other possible external magnetic fields.
The Ernst black hole, or rotating Ernst black hole has a magnetic field in the very structure of the spacetime, and as well as the Reissner-Nordström, Kerr and Kerr-Newman, etc, it is expected all of these have different kinds of fields as background fields via metric tensor. This occurs mathematically (look at the metric tensor) and physically I think the picture should be seen in two fashions, one, the event horizon, once formed has the measurable properties for the external observer associated with the presence of the fields, second, thinking in the black hole as the physical remnant of a star or so-like object, the spacetime if magnetic in the original star spacetime (Ernst-like) will remain so after the star collapse, being again the event horizon responsible for some of the associated properties of this spacetime like the thermodynamics descending from it.
A: First of all your first point, namely that

the extreme forces compressing the atoms destabilizes and destroys their structure to the point that matter inside a black hole cannot even manifest the emergent properties of combining quarks.

is somewhat incorrect. Matter entering black hole would encounter such "extreme forces" only after it crosses horizon, while before that the effects of gravity would not be that extreme. In fact, the larger the black hole, the less severe conditions at the horizon crossing would be. Moreover, there are several variations of general relativity where the matter falling into a black hole would encounter some sort of "Big Bounce" rather than "Big Crunch". But precise fate of the matter under horizon does not, uh, matter because of your second point, which is essentially correct.
Specifically, as the black hole forms in the collapse (or grows by absorbing matter) most of parameters with which we can characterize the system (stationary fields, including magnetic, various types of radiation) disappear as black hole stabilizes, with their sources being absorbed by black hole or radiated away. What parameters are left is given by the no-hair theorem which states that isolated black hole sheds through radiation all those characteristic that radiation can remove. Radiation of spin $s$ field could change the system's multipole moment $l$, but only if $l\ge s$. Since gravitons have spin $s = 2$, moments with $l = 0$ and $l=1$ survive and so the black hole is characterized  by mass ($l=0$ moment) and angular momentum ($l=1$ moment). Electromagnetic field has spin $s=1$, and so a black hole formed from electromagnetically interacting matter can also have electric charge $Q$ ($l=0$ moment of EM field), but not higher moments and so a black hole could not have a magnetic moment. Note, that since stars are to a high degree of precision electrically neutral, charges of real black holes would be rather small.
Another possible $l=0$ electromagnetic moment potentially observable in black holes is the monopole charge from hypothesized magnetic monopoles. While many grand unification theories predict this kind of particle, so far there is no experimental evidence that magnetic monopoles exist. But if they do, black holes can carry magnetic monopole charge.
Finally, even without electric or magnetic charges black holes can still 'react strongly to the presence of an external magnetic field'. If magnetic field lines (from external currents) are threaded through the rotating black hole then an electric potential difference can occur, possibly large enough for production of electron-positron pairs. This would allow extraction of energy from the black hole. This Blandford–Znajek process is most likely the mechanism with which quazars are powered.
